It follows easily by use of [1]} that the inner \(\alpha \) -Riesz equilibrium measure \(\gamma _A\)  is unique (if exists). Theorem REF  below provides a number of equivalent conditions, each of which is necessary and sufficient for the existence of this \(\gamma _A\) .
The approach applied is based on a close interaction between the concept of inner equilibrium measure and that of inner balayage, described by means of equality (REF ) with the Kelvin transformation involved.
